Discrete Rotations on the Triangular Plane

There are various geometric transformations, e.g., translations, rotations, which are always bijections in the Euclidean space. Their digital counterpart, i.e., their digitized variants are defined on discrete grids, since most of our pictures are digital nowadays. Usually, these digital versions of the transformations have different properties than the original continuous variants have. Rotations are bijective on the Euclidean plane, but in many cases they are not injective and not surjective on digital grids. Since these transformations play an important role in image processing and in image manipulation, it is important to discover their properties. Neighborhood motion maps are tools to analyze digital transformations, e.g., rotations by local bijectivity point of view. In this thesis we show digitized rotations of a pixel and its 12-neighbors on the triangular grid. In particular, different rotation centers are considered with respect to the corresponding main pixel, e.g. edge midpoints and corner points. Angles of all locally bijective and non-bijective rotations are described in details. It is also shown that the triangular grid shows better performance in some cases than the square grid regarding the number of lost pixels in the neighborhood motion map. We also compare the bijectivity of the digitized rotations for the closest neighbors in all three regular grids. Rotations for every integer degree are studied for rotation centers at the corner, edgemidpoint and at the center of the pixels. The experiment proves that, when the center of rotation is the center or the corner of a main pixel, then, the triangular grid and the square grid have a better behaviour compared to the hexagonal grid. Keywords: Digital rotations, Discrete motions, Non-traditional grids, Neighborhood motion maps, Lost pixels, Quality of images.